Theorem fnbr

Description: The first argument of binary relation on a function belongs to the function's domain. (Contributed by NM, 7-May-2004)

	Ref	Expression
Assertion	fnbr	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 𝐹 𝐶 ) → 𝐵 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		fnrel	⊢ ( 𝐹 Fn 𝐴 → Rel 𝐹 )
2		releldm	⊢ ( ( Rel 𝐹 ∧ 𝐵 𝐹 𝐶 ) → 𝐵 ∈ dom 𝐹 )
3	1 2	sylan	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 𝐹 𝐶 ) → 𝐵 ∈ dom 𝐹 )
4		fndm	⊢ ( 𝐹 Fn 𝐴 → dom 𝐹 = 𝐴 )
5	4	eleq2d	⊢ ( 𝐹 Fn 𝐴 → ( 𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝐴 ) )
6	5	biimpa	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ∈ dom 𝐹 ) → 𝐵 ∈ 𝐴 )
7	3 6	syldan	⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 𝐹 𝐶 ) → 𝐵 ∈ 𝐴 )